4 Fractional Stationarity

[554.1.1]This section investigates the condition of invariance or stationarity for the induced ultralong time dynamics Sϖt*. [554.1.2]Invariance of a measure ν on G under the induced dynamics Sϖt* is defined as usual (see (3)) by requiring that

for t>0 and B⊂G. [554.1.3]For 0<ϖ<1 (22) may be called the condition of fractional invariance or fractional stationarity. [554.1.4]Using (2) the invariance condition becomes

[page 555, §0]   for t>0 where Aϖ is the infinitesimal generator of the semigroup Sϖt*. [555.0.1]For ϖ=1 the relation (21) implies A1⁢ν⁢B,t=-d⁢ν⁢B,t/d⁢t=0, and thus in this case invariant measures conserve volumes in phase space as usual. [555.0.2]A very different situation arises for ϖ<1.

[555.1.1]For 0<ϖ<1 the infinitesimal generators of the stable convolution semigroup Sϖt* are obtained [26] by evaluating the generalized function s+-ϖ-1 [29] on the time translation group Ts

where c+>0 is a constant. [555.1.2]Comparing (24) with the Balakrishnan algorithm [30, 31, 32] for fractional powers of the generator of a semigroup Tt

shows that if A=-d/d⁢t denotes the infinitesimal generator of the original time evolution Tt then Aϖ=-Aϖ is the infinitesimal generator of the induced time evolution Sϖt*. [555.1.3]For 0<ϖ<1 the generators Aϖ for Sϖt* are fractional time derivatives [15, 31, 29]. [555.1.4]The differential form (23) of the fractional invariance condition for ν becomes

for t>0 which was first derived in [18, 19]. [555.1.5]Its solution is

for t>0 with C0 a constant. [555.1.6]This shows that ν⁢B for a fractional stationary dynamical state is not constant. [555.1.7]Fractional stationarity or fractional invariance of a measure ν implies that phase space volumes ν⁢B shrink with time. [555.1.8]Thus fractional dynamics is dissipative. [555.1.9]More generally (26) reads Aϖ⁢ν⁢B,t=δ⁢t with solution ν⁢B,t=C0⁢t+ϖ-1 for t≥0 in the sense of distributions. [555.1.10]The stationary solution with ϖ=1 has a jump discontinuity at t=0, and is not simply constant.

[555.2.1]The transition from an original invariant measure μ on Γ to a fractional invariant measure ν on a subset G of measure μ⁢G=0 may be called stationarity breaking. [555.2.2]It occurs spontaneously in the sense that it is generated by the dynamics itself. [555.2.3]Stationarity breaking implies ergodicity breaking, and thus the ultralong time limit is a possible scenario for ergodicity breaking in ergodic theory.

[555.3.1]The present paper has shown that the use of fractional time derivatives in physics is not only justified, but arises generically for induced dynamics in the ultralong time limit. [555.3.2]This mathematical result applies to many physical situations. [555.3.3]In the simplest case the resulting fractional differential equation (26) defines fractional stationarity which provides the dynamical basis for the anequilibrium concept [18]. [555.3.4]Recently fractional random walks were discussed [8] and solved [10] in the continuum limit.

[page 556, §0]   ACKNOWLEDGEMENT : The author is grateful to Norges Forskningsrad (Nr.: 424.94 / 004 B) for financial support.